TPTP Problem File: SEU824^3.p
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%------------------------------------------------------------------------------
% File : SEU824^3 : TPTP v9.0.0. Released v3.7.0.
% Domain : Set Theory
% Problem : Ordinals
% Version : Especial > Reduced > Especial.
% English : Just the axioms are provided, with $false as the conjecture.
% Refs : [Bro09] Brown (2008), Email to G. Sutcliffe
% Source : [Bro09]
% Names : ZFC326gc [Bro08]
% Status : CounterSatisfiable
% Rating : 1.00 v3.7.0
% Syntax : Number of formulae : 409 ( 54 unt; 64 typ; 4 def)
% Number of atoms : 1105 ( 150 equ; 0 cnn)
% Maximal formula atoms : 26 ( 3 avg)
% Number of connectives : 4238 ( 80 ~; 17 |; 60 &;3313 @)
% ( 17 <=>; 751 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 9 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 195 ( 195 >; 0 *; 0 +; 0 <<)
% Number of symbols : 67 ( 64 usr; 4 con; 0-4 aty)
% Number of variables : 1185 ( 77 ^;1063 !; 45 ?;1185 :)
% SPC : TH0_CSA_EQU_NAR
% Comments : http://mathgate.info/detsetitem.php?id=521
%------------------------------------------------------------------------------
thf(in_type,type,
in: $i > $i > $o ).
thf(exu_type,type,
exu: ( $i > $o ) > $o ).
thf(exu,definition,
( exu
= ( ^ [Xphi: $i > $o] :
? [Xx: $i] :
( ( Xphi @ Xx )
& ! [Xy: $i] :
( ( Xphi @ Xy )
=> ( Xx = Xy ) ) ) ) ) ).
thf(setextAx,axiom,
! [A: $i,B: $i] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
<=> ( in @ Xx @ B ) )
=> ( A = B ) ) ).
thf(emptyset_type,type,
emptyset: $i ).
thf(emptysetAx,axiom,
! [Xx: $i] :
~ ( in @ Xx @ emptyset ) ).
thf(setadjoin_type,type,
setadjoin: $i > $i > $i ).
thf(setadjoinAx,axiom,
! [Xx: $i,A: $i,Xy: $i] :
( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
<=> ( ( Xy = Xx )
| ( in @ Xy @ A ) ) ) ).
thf(powerset_type,type,
powerset: $i > $i ).
thf(powersetAx,axiom,
! [A: $i,B: $i] :
( ( in @ B @ ( powerset @ A ) )
<=> ! [Xx: $i] :
( ( in @ Xx @ B )
=> ( in @ Xx @ A ) ) ) ).
thf(setunion_type,type,
setunion: $i > $i ).
thf(setunionAx,axiom,
! [A: $i,Xx: $i] :
( ( in @ Xx @ ( setunion @ A ) )
<=> ? [B: $i] :
( ( in @ Xx @ B )
& ( in @ B @ A ) ) ) ).
thf(omega_type,type,
omega: $i ).
thf(omega0Ax,axiom,
in @ emptyset @ omega ).
thf(omegaSAx,axiom,
! [Xx: $i] :
( ( in @ Xx @ omega )
=> ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ).
thf(omegaIndAx,axiom,
! [A: $i] :
( ( ( in @ emptyset @ A )
& ! [Xx: $i] :
( ( ( in @ Xx @ omega )
& ( in @ Xx @ A ) )
=> ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
=> ! [Xx: $i] :
( ( in @ Xx @ omega )
=> ( in @ Xx @ A ) ) ) ).
thf(replAx,axiom,
! [Xphi: $i > $i > $o,A: $i] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( exu
@ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
=> ? [B: $i] :
! [Xx: $i] :
( ( in @ Xx @ B )
<=> ? [Xy: $i] :
( ( in @ Xy @ A )
& ( Xphi @ Xy @ Xx ) ) ) ) ).
thf(foundationAx,axiom,
! [A: $i] :
( ? [Xx: $i] : ( in @ Xx @ A )
=> ? [B: $i] :
( ( in @ B @ A )
& ~ ? [Xx: $i] :
( ( in @ Xx @ B )
& ( in @ Xx @ A ) ) ) ) ).
thf(wellorderingAx,axiom,
! [A: $i] :
? [B: $i] :
( ! [C: $i] :
( ( in @ C @ B )
=> ! [Xx: $i] :
( ( in @ Xx @ C )
=> ( in @ Xx @ A ) ) )
& ! [Xx: $i,Xy: $i] :
( ( ( in @ Xx @ A )
& ( in @ Xy @ A ) )
=> ( ! [C: $i] :
( ( in @ C @ B )
=> ( ( in @ Xx @ C )
<=> ( in @ Xy @ C ) ) )
=> ( Xx = Xy ) ) )
& ! [C: $i,D: $i] :
( ( ( in @ C @ B )
& ( in @ D @ B ) )
=> ( ! [Xx: $i] :
( ( in @ Xx @ C )
=> ( in @ Xx @ D ) )
| ! [Xx: $i] :
( ( in @ Xx @ D )
=> ( in @ Xx @ C ) ) ) )
& ! [C: $i] :
( ( ! [Xx: $i] :
( ( in @ Xx @ C )
=> ( in @ Xx @ A ) )
& ? [Xx: $i] : ( in @ Xx @ C ) )
=> ? [D: $i,Xx: $i] :
( ( in @ D @ B )
& ( in @ Xx @ C )
& ~ ? [Xy: $i] :
( ( in @ Xy @ D )
& ( in @ Xy @ C ) )
& ! [E: $i] :
( ( in @ E @ B )
=> ( ! [Xy: $i] :
( ( in @ Xy @ E )
=> ( in @ Xy @ D ) )
| ( in @ Xx @ E ) ) ) ) ) ) ).
thf(descr_type,type,
descr: ( $i > $o ) > $i ).
thf(descrp,axiom,
! [Xphi: $i > $o] :
( ( exu
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
=> ( Xphi
@ ( descr
@ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
thf(dsetconstr_type,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(dsetconstrI,axiom,
! [A: $i,Xphi: $i > $o,Xx: $i] :
( ( in @ Xx @ A )
=> ( ( Xphi @ Xx )
=> ( in @ Xx
@ ( dsetconstr @ A
@ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ).
thf(dsetconstrEL,axiom,
! [A: $i,Xphi: $i > $o,Xx: $i] :
( ( in @ Xx
@ ( dsetconstr @ A
@ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
=> ( in @ Xx @ A ) ) ).
thf(dsetconstrER,axiom,
! [A: $i,Xphi: $i > $o,Xx: $i] :
( ( in @ Xx
@ ( dsetconstr @ A
@ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
=> ( Xphi @ Xx ) ) ).
thf(exuE1,axiom,
! [Xphi: $i > $o] :
( ( exu
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
=> ? [Xx: $i] :
( ( Xphi @ Xx )
& ! [Xy: $i] :
( ( Xphi @ Xy )
=> ( Xx = Xy ) ) ) ) ).
thf(prop2set_type,type,
prop2set: $o > $i ).
thf(prop2setE,axiom,
! [Xphi: $o,Xx: $i] :
( ( in @ Xx @ ( prop2set @ Xphi ) )
=> Xphi ) ).
thf(emptysetE,axiom,
! [Xx: $i] :
( ( in @ Xx @ emptyset )
=> ! [Xphi: $o] : Xphi ) ).
thf(emptysetimpfalse,axiom,
! [Xx: $i] :
( ( in @ Xx @ emptyset )
=> $false ) ).
thf(notinemptyset,axiom,
! [Xx: $i] :
~ ( in @ Xx @ emptyset ) ).
thf(exuE3e,axiom,
! [Xphi: $i > $o] :
( ( exu
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
=> ? [Xx: $i] : ( Xphi @ Xx ) ) ).
thf(setext,axiom,
! [A: $i,B: $i] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ Xx @ B ) )
=> ( ! [Xx: $i] :
( ( in @ Xx @ B )
=> ( in @ Xx @ A ) )
=> ( A = B ) ) ) ).
thf(emptyI,axiom,
! [A: $i] :
( ! [Xx: $i] :
~ ( in @ Xx @ A )
=> ( A = emptyset ) ) ).
thf(noeltsimpempty,axiom,
! [A: $i] :
( ! [Xx: $i] :
~ ( in @ Xx @ A )
=> ( A = emptyset ) ) ).
thf(setbeta,axiom,
! [A: $i,Xphi: $i > $o,Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx
@ ( dsetconstr @ A
@ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
<=> ( Xphi @ Xx ) ) ) ).
thf(nonempty_type,type,
nonempty: $i > $o ).
thf(nonemptyE1,axiom,
! [A: $i] :
( ( nonempty @ A )
=> ? [Xx: $i] : ( in @ Xx @ A ) ) ).
thf(nonemptyI,axiom,
! [A: $i,Xphi: $i > $o,Xx: $i] :
( ( in @ Xx @ A )
=> ( ( Xphi @ Xx )
=> ( nonempty
@ ( dsetconstr @ A
@ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ).
thf(nonemptyI1,axiom,
! [A: $i] :
( ? [Xx: $i] : ( in @ Xx @ A )
=> ( nonempty @ A ) ) ).
thf(setadjoinIL,axiom,
! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ).
thf(emptyinunitempty,axiom,
in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ).
thf(setadjoinIR,axiom,
! [Xx: $i,A: $i,Xy: $i] :
( ( in @ Xy @ A )
=> ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ).
thf(setadjoinE,axiom,
! [Xx: $i,A: $i,Xy: $i] :
( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
=> ! [Xphi: $o] :
( ( ( Xy = Xx )
=> Xphi )
=> ( ( ( in @ Xy @ A )
=> Xphi )
=> Xphi ) ) ) ).
thf(setadjoinOr,axiom,
! [Xx: $i,A: $i,Xy: $i] :
( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
=> ( ( Xy = Xx )
| ( in @ Xy @ A ) ) ) ).
thf(setoftrueEq,axiom,
! [A: $i] :
( ( dsetconstr @ A
@ ^ [Xx: $i] : $true )
= A ) ).
thf(powersetI,axiom,
! [A: $i,B: $i] :
( ! [Xx: $i] :
( ( in @ Xx @ B )
=> ( in @ Xx @ A ) )
=> ( in @ B @ ( powerset @ A ) ) ) ).
thf(emptyinPowerset,axiom,
! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ).
thf(emptyInPowerset,axiom,
! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ).
thf(powersetE,axiom,
! [A: $i,B: $i,Xx: $i] :
( ( in @ B @ ( powerset @ A ) )
=> ( ( in @ Xx @ B )
=> ( in @ Xx @ A ) ) ) ).
thf(setunionI,axiom,
! [A: $i,Xx: $i,B: $i] :
( ( in @ Xx @ B )
=> ( ( in @ B @ A )
=> ( in @ Xx @ ( setunion @ A ) ) ) ) ).
thf(setunionE,axiom,
! [A: $i,Xx: $i] :
( ( in @ Xx @ ( setunion @ A ) )
=> ! [Xphi: $o] :
( ! [B: $i] :
( ( in @ Xx @ B )
=> ( ( in @ B @ A )
=> Xphi ) )
=> Xphi ) ) ).
thf(subPowSU,axiom,
! [A: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ).
thf(exuE2,axiom,
! [Xphi: $i > $o] :
( ( exu
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
=> ? [Xx: $i] :
! [Xy: $i] :
( ( Xphi @ Xy )
<=> ( Xy = Xx ) ) ) ).
thf(nonemptyImpWitness,axiom,
! [A: $i] :
( ( nonempty @ A )
=> ? [Xx: $i] :
( ( in @ Xx @ A )
& $true ) ) ).
thf(uniqinunit,axiom,
! [Xx: $i,Xy: $i] :
( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
=> ( Xx = Xy ) ) ).
thf(notinsingleton,axiom,
! [Xx: $i,Xy: $i] :
( ( Xx != Xy )
=> ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ).
thf(eqinunit,axiom,
! [Xx: $i,Xy: $i] :
( ( Xx = Xy )
=> ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ).
thf(singletonsswitch,axiom,
! [Xx: $i,Xy: $i] :
( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
=> ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ).
thf(upairsetE,axiom,
! [Xx: $i,Xy: $i,Xz: $i] :
( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
=> ( ( Xz = Xx )
| ( Xz = Xy ) ) ) ).
thf(upairsetIL,axiom,
! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ).
thf(upairsetIR,axiom,
! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ).
thf(emptyE1,axiom,
! [A: $i,Xphi: $i > $o] :
( ? [Xx: $i] :
( ( in @ Xx @ A )
& ( Xphi @ Xx ) )
=> ( ( ( dsetconstr @ A
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
= emptyset )
=> $false ) ) ).
thf(vacuousDall,axiom,
! [Xphi: $i > $o,Xx: $i] :
( ( in @ Xx @ emptyset )
=> ( Xphi @ Xx ) ) ).
thf(quantDeMorgan1,axiom,
! [A: $i,Xphi: $i > $o] :
( ~ ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( Xphi @ Xx ) )
=> ? [Xx: $i] :
( ( in @ Xx @ A )
& ~ ( Xphi @ Xx ) ) ) ).
thf(quantDeMorgan2,axiom,
! [A: $i,Xphi: $i > $o] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ~ ( Xphi @ Xx ) )
=> ~ ? [Xx: $i] :
( ( in @ Xx @ A )
& ( Xphi @ Xx ) ) ) ).
thf(quantDeMorgan3,axiom,
! [A: $i,Xphi: $i > $o] :
( ~ ? [Xx: $i] :
( ( in @ Xx @ A )
& ( Xphi @ Xx ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ~ ( Xphi @ Xx ) ) ) ).
thf(quantDeMorgan4,axiom,
! [A: $i,Xphi: $i > $o] :
( ? [Xx: $i] :
( ( in @ Xx @ A )
& ~ ( Xphi @ Xx ) )
=> ~ ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( Xphi @ Xx ) ) ) ).
thf(prop2setI,axiom,
! [Xphi: $o] :
( Xphi
=> ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ).
thf(set2prop_type,type,
set2prop: $i > $o ).
thf(prop2set2propI,axiom,
! [Xphi: $o] :
( Xphi
=> ( set2prop @ ( prop2set @ Xphi ) ) ) ).
thf(notdexE,axiom,
! [A: $i,Xphi: $i > $o] :
( ~ ? [Xx: $i] :
( ( in @ Xx @ A )
& ( Xphi @ Xx ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ~ ( Xphi @ Xx ) ) ) ).
thf(notdallE,axiom,
! [A: $i,Xphi: $i > $o] :
( ~ ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( Xphi @ Xx ) )
=> ? [Xx: $i] :
( ( in @ Xx @ A )
& ~ ( Xphi @ Xx ) ) ) ).
thf(exuI1,axiom,
! [Xphi: $i > $o] :
( ? [Xx: $i] :
( ( Xphi @ Xx )
& ! [Xy: $i] :
( ( Xphi @ Xy )
=> ( Xx = Xy ) ) )
=> ( exu
@ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ).
thf(exuI3,axiom,
! [Xphi: $i > $o] :
( ? [Xx: $i] : ( Xphi @ Xx )
=> ( ! [Xx: $i,Xy: $i] :
( ( Xphi @ Xx )
=> ( ( Xphi @ Xy )
=> ( Xx = Xy ) ) )
=> ( exu
@ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
thf(exuI2,axiom,
! [Xphi: $i > $o] :
( ? [Xx: $i] :
! [Xy: $i] :
( ( Xphi @ Xy )
<=> ( Xy = Xx ) )
=> ( exu
@ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ).
thf(inCongP,axiom,
! [A: $i,B: $i] :
( ( A = B )
=> ! [Xx: $i,Xy: $i] :
( ( Xx = Xy )
=> ( ( in @ Xx @ A )
=> ( in @ Xy @ B ) ) ) ) ).
thf(in__Cong,axiom,
! [A: $i,B: $i] :
( ( A = B )
=> ! [Xx: $i,Xy: $i] :
( ( Xx = Xy )
=> ( ( in @ Xx @ A )
<=> ( in @ Xy @ B ) ) ) ) ).
thf(exuE3u,axiom,
! [Xphi: $i > $o] :
( ( exu
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
=> ! [Xx: $i,Xy: $i] :
( ( Xphi @ Xx )
=> ( ( Xphi @ Xy )
=> ( Xx = Xy ) ) ) ) ).
thf(exu__Cong,axiom,
! [Xphi: $i > $o,Xpsi: $i > $o] :
( ! [Xx: $i,Xy: $i] :
( ( Xx = Xy )
=> ( ( Xphi @ Xx )
<=> ( Xpsi @ Xy ) ) )
=> ( ( exu
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
<=> ( exu
@ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ).
thf(emptyset__Cong,axiom,
emptyset = emptyset ).
thf(setadjoin__Cong,axiom,
! [Xx: $i,Xy: $i] :
( ( Xx = Xy )
=> ! [Xz: $i,Xu: $i] :
( ( Xz = Xu )
=> ( ( setadjoin @ Xx @ Xz )
= ( setadjoin @ Xy @ Xu ) ) ) ) ).
thf(powerset__Cong,axiom,
! [A: $i,B: $i] :
( ( A = B )
=> ( ( powerset @ A )
= ( powerset @ B ) ) ) ).
thf(setunion__Cong,axiom,
! [A: $i,B: $i] :
( ( A = B )
=> ( ( setunion @ A )
= ( setunion @ B ) ) ) ).
thf(omega__Cong,axiom,
omega = omega ).
thf(exuEu,axiom,
! [Xphi: $i > $o] :
( ( exu
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
=> ! [Xx: $i,Xy: $i] :
( ( Xphi @ Xx )
=> ( ( Xphi @ Xy )
=> ( Xx = Xy ) ) ) ) ).
thf(descr__Cong,axiom,
! [Xphi: $i > $o,Xpsi: $i > $o] :
( ! [Xx: $i,Xy: $i] :
( ( Xx = Xy )
=> ( ( Xphi @ Xx )
<=> ( Xpsi @ Xy ) ) )
=> ( ( exu
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
=> ( ( exu
@ ^ [Xx: $i] : ( Xpsi @ Xx ) )
=> ( ( descr
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
= ( descr
@ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
thf(dsetconstr__Cong,axiom,
! [A: $i,B: $i] :
( ( A = B )
=> ! [Xphi: $i > $o,Xpsi: $i > $o] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( ( Xx = Xy )
=> ( ( Xphi @ Xx )
<=> ( Xpsi @ Xy ) ) ) ) )
=> ( ( dsetconstr @ A
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
= ( dsetconstr @ B
@ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ).
thf(subset_type,type,
subset: $i > $i > $o ).
thf(disjoint_type,type,
disjoint: $i > $i > $o ).
thf(setsmeet_type,type,
setsmeet: $i > $i > $o ).
thf(subsetI1,axiom,
! [A: $i,B: $i] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ Xx @ B ) )
=> ( subset @ A @ B ) ) ).
thf(eqimpsubset2,axiom,
! [A: $i,B: $i] :
( ( A = B )
=> ( subset @ B @ A ) ) ).
thf(eqimpsubset1,axiom,
! [A: $i,B: $i] :
( ( A = B )
=> ( subset @ A @ B ) ) ).
thf(subsetI2,axiom,
! [A: $i,B: $i] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ Xx @ B ) )
=> ( subset @ A @ B ) ) ).
thf(emptysetsubset,axiom,
! [A: $i] : ( subset @ emptyset @ A ) ).
thf(subsetE,axiom,
! [A: $i,B: $i,Xx: $i] :
( ( subset @ A @ B )
=> ( ( in @ Xx @ A )
=> ( in @ Xx @ B ) ) ) ).
thf(subsetE2,axiom,
! [A: $i,B: $i,Xx: $i] :
( ( subset @ A @ B )
=> ( ~ ( in @ Xx @ B )
=> ~ ( in @ Xx @ A ) ) ) ).
thf(notsubsetI,axiom,
! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ( ~ ( in @ Xx @ B )
=> ~ ( subset @ A @ B ) ) ) ).
thf(notequalI1,axiom,
! [A: $i,B: $i] :
( ~ ( subset @ A @ B )
=> ( A != B ) ) ).
thf(notequalI2,axiom,
! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ( ~ ( in @ Xx @ B )
=> ( A != B ) ) ) ).
thf(subsetRefl,axiom,
! [A: $i] : ( subset @ A @ A ) ).
thf(subsetTrans,axiom,
! [A: $i,B: $i,C: $i] :
( ( subset @ A @ B )
=> ( ( subset @ B @ C )
=> ( subset @ A @ C ) ) ) ).
thf(setadjoinSub,axiom,
! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ).
thf(setadjoinSub2,axiom,
! [A: $i,Xx: $i,B: $i] :
( ( subset @ A @ B )
=> ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ).
thf(subset2powerset,axiom,
! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( in @ A @ ( powerset @ B ) ) ) ).
thf(setextsub,axiom,
! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( ( subset @ B @ A )
=> ( A = B ) ) ) ).
thf(subsetemptysetimpeq,axiom,
! [A: $i] :
( ( subset @ A @ emptyset )
=> ( A = emptyset ) ) ).
thf(powersetI1,axiom,
! [A: $i,B: $i] :
( ( subset @ B @ A )
=> ( in @ B @ ( powerset @ A ) ) ) ).
thf(powersetE1,axiom,
! [A: $i,B: $i] :
( ( in @ B @ ( powerset @ A ) )
=> ( subset @ B @ A ) ) ).
thf(inPowerset,axiom,
! [A: $i] : ( in @ A @ ( powerset @ A ) ) ).
thf(powersetsubset,axiom,
! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ).
thf(sepInPowerset,axiom,
! [A: $i,Xphi: $i > $o] :
( in
@ ( dsetconstr @ A
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
@ ( powerset @ A ) ) ).
thf(sepSubset,axiom,
! [A: $i,Xphi: $i > $o] :
( subset
@ ( dsetconstr @ A
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
@ A ) ).
thf(binunion_type,type,
binunion: $i > $i > $i ).
thf(binunionIL,axiom,
! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ Xx @ ( binunion @ A @ B ) ) ) ).
thf(upairset2IR,axiom,
! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ).
thf(binunionIR,axiom,
! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ B )
=> ( in @ Xx @ ( binunion @ A @ B ) ) ) ).
thf(binunionEcases,axiom,
! [A: $i,B: $i,Xx: $i,Xphi: $o] :
( ( in @ Xx @ ( binunion @ A @ B ) )
=> ( ( ( in @ Xx @ A )
=> Xphi )
=> ( ( ( in @ Xx @ B )
=> Xphi )
=> Xphi ) ) ) ).
thf(binunionE,axiom,
! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ ( binunion @ A @ B ) )
=> ( ( in @ Xx @ A )
| ( in @ Xx @ B ) ) ) ).
thf(binunionLsub,axiom,
! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ).
thf(binunionRsub,axiom,
! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ).
thf(binintersect_type,type,
binintersect: $i > $i > $i ).
thf(binintersectI,axiom,
! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ B )
=> ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ).
thf(binintersectSubset5,axiom,
! [A: $i,B: $i,C: $i] :
( ( subset @ C @ A )
=> ( ( subset @ C @ B )
=> ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ).
thf(binintersectEL,axiom,
! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ ( binintersect @ A @ B ) )
=> ( in @ Xx @ A ) ) ).
thf(binintersectLsub,axiom,
! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ).
thf(binintersectSubset2,axiom,
! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( ( binintersect @ A @ B )
= A ) ) ).
thf(binintersectSubset3,axiom,
! [A: $i,B: $i] :
( ( ( binintersect @ A @ B )
= B )
=> ( subset @ B @ A ) ) ).
thf(binintersectER,axiom,
! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ ( binintersect @ A @ B ) )
=> ( in @ Xx @ B ) ) ).
thf(disjointsetsI1,axiom,
! [A: $i,B: $i] :
( ~ ? [Xx: $i] :
( ( in @ Xx @ A )
& ( in @ Xx @ B ) )
=> ( ( binintersect @ A @ B )
= emptyset ) ) ).
thf(binintersectRsub,axiom,
! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ).
thf(binintersectSubset4,axiom,
! [A: $i,B: $i] :
( ( subset @ B @ A )
=> ( ( binintersect @ A @ B )
= B ) ) ).
thf(binintersectSubset1,axiom,
! [A: $i,B: $i] :
( ( ( binintersect @ A @ B )
= A )
=> ( subset @ A @ B ) ) ).
thf(bs114d,axiom,
! [A: $i,B: $i,C: $i] :
( ( binintersect @ A @ ( binunion @ B @ C ) )
= ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ).
thf(regular_type,type,
regular: $i > $o ).
thf(setminus_type,type,
setminus: $i > $i > $i ).
thf(setminusI,axiom,
! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ( ~ ( in @ Xx @ B )
=> ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ).
thf(setminusEL,axiom,
! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ ( setminus @ A @ B ) )
=> ( in @ Xx @ A ) ) ).
thf(setminusER,axiom,
! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ ( setminus @ A @ B ) )
=> ~ ( in @ Xx @ B ) ) ).
thf(setminusSubset2,axiom,
! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( ( setminus @ A @ B )
= emptyset ) ) ).
thf(setminusERneg,axiom,
! [A: $i,B: $i,Xx: $i] :
( ~ ( in @ Xx @ ( setminus @ A @ B ) )
=> ( ( in @ Xx @ A )
=> ( in @ Xx @ B ) ) ) ).
thf(setminusELneg,axiom,
! [A: $i,B: $i,Xx: $i] :
( ~ ( in @ Xx @ ( setminus @ A @ B ) )
=> ( ~ ( in @ Xx @ B )
=> ~ ( in @ Xx @ A ) ) ) ).
thf(setminusILneg,axiom,
! [A: $i,B: $i,Xx: $i] :
( ~ ( in @ Xx @ A )
=> ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ).
thf(setminusIRneg,axiom,
! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ B )
=> ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ).
thf(setminusLsub,axiom,
! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ).
thf(setminusSubset1,axiom,
! [A: $i,B: $i] :
( ( ( setminus @ A @ B )
= emptyset )
=> ( subset @ A @ B ) ) ).
thf(symdiff_type,type,
symdiff: $i > $i > $i ).
thf(symdiffE,axiom,
! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ ( symdiff @ A @ B ) )
=> ! [Xphi: $o] :
( ( ( in @ Xx @ A )
=> ( ~ ( in @ Xx @ B )
=> Xphi ) )
=> ( ( ~ ( in @ Xx @ A )
=> ( ( in @ Xx @ B )
=> Xphi ) )
=> Xphi ) ) ) ).
thf(symdiffI1,axiom,
! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ( ~ ( in @ Xx @ B )
=> ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ).
thf(symdiffI2,axiom,
! [A: $i,B: $i,Xx: $i] :
( ~ ( in @ Xx @ A )
=> ( ( in @ Xx @ B )
=> ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ).
thf(symdiffIneg1,axiom,
! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ B )
=> ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ).
thf(symdiffIneg2,axiom,
! [A: $i,B: $i,Xx: $i] :
( ~ ( in @ Xx @ A )
=> ( ~ ( in @ Xx @ B )
=> ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ).
thf(iskpair_type,type,
iskpair: $i > $o ).
thf(secondinupair,axiom,
! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ).
thf(setukpairIL,axiom,
! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ).
thf(setukpairIR,axiom,
! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ).
thf(kpairiskpair,axiom,
! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ).
thf(kpair_type,type,
kpair: $i > $i > $i ).
thf(kpairp,axiom,
! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ).
thf(cartprod_type,type,
cartprod: $i > $i > $i ).
thf(singletonsubset,axiom,
! [A: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ).
thf(singletoninpowerset,axiom,
! [A: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ).
thf(singletoninpowunion,axiom,
! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ).
thf(upairset2E,axiom,
! [Xx: $i,Xy: $i,Xz: $i] :
( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
=> ( ( Xz = Xx )
| ( Xz = Xy ) ) ) ).
thf(upairsubunion,axiom,
! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ).
thf(upairinpowunion,axiom,
! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ).
thf(ubforcartprodlem1,axiom,
! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ).
thf(ubforcartprodlem2,axiom,
! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
thf(ubforcartprodlem3,axiom,
! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
thf(cartprodpairin,axiom,
! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ).
thf(cartprodmempair1,axiom,
! [A: $i,B: $i,Xu: $i] :
( ( in @ Xu @ ( cartprod @ A @ B ) )
=> ? [Xx: $i] :
( ( in @ Xx @ A )
& ? [Xy: $i] :
( ( in @ Xy @ B )
& ( Xu
= ( kpair @ Xx @ Xy ) ) ) ) ) ).
thf(cartprodmempair,axiom,
! [A: $i,B: $i,Xu: $i] :
( ( in @ Xu @ ( cartprod @ A @ B ) )
=> ( iskpair @ Xu ) ) ).
thf(setunionE2,axiom,
! [A: $i,Xx: $i] :
( ( in @ Xx @ ( setunion @ A ) )
=> ? [X: $i] :
( ( in @ X @ A )
& ( in @ Xx @ X ) ) ) ).
thf(setunionsingleton1,axiom,
! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ).
thf(setunionsingleton2,axiom,
! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ).
thf(setunionsingleton,axiom,
! [Xx: $i] :
( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
= Xx ) ).
thf(singleton_type,type,
singleton: $i > $o ).
thf(singletonprop,axiom,
! [A: $i,Xphi: $i > $o] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ( Xphi @ Xx )
=> ( ( Xphi @ Xy )
=> ( Xx = Xy ) ) ) ) )
=> ( ? [Xx: $i] :
( ( in @ Xx @ A )
& ( Xphi @ Xx ) )
=> ( singleton
@ ( dsetconstr @ A
@ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
thf(ex1_type,type,
ex1: $i > ( $i > $o ) > $o ).
thf(ex1E1,axiom,
! [A: $i,Xphi: $i > $o] :
( ( ex1 @ A
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
=> ? [Xx: $i] :
( ( in @ Xx @ A )
& ( Xphi @ Xx ) ) ) ).
thf(ex1I,axiom,
! [A: $i,Xphi: $i > $o,Xx: $i] :
( ( in @ Xx @ A )
=> ( ( Xphi @ Xx )
=> ( ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ( Xphi @ Xy )
=> ( Xy = Xx ) ) )
=> ( ex1 @ A
@ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ).
thf(ex1I2,axiom,
! [A: $i,Xphi: $i > $o] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ( Xphi @ Xx )
=> ( ( Xphi @ Xy )
=> ( Xx = Xy ) ) ) ) )
=> ( ? [Xx: $i] :
( ( in @ Xx @ A )
& ( Xphi @ Xx ) )
=> ( ex1 @ A
@ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
thf(singletonsuniq,axiom,
! [Xx: $i,Xy: $i] :
( ( ( setadjoin @ Xx @ emptyset )
= ( setadjoin @ Xy @ emptyset ) )
=> ( Xx = Xy ) ) ).
thf(atmost1p_type,type,
atmost1p: $i > $o ).
thf(atleast2p_type,type,
atleast2p: $i > $o ).
thf(atmost2p_type,type,
atmost2p: $i > $o ).
thf(upairsetp_type,type,
upairsetp: $i > $o ).
thf(setukpairinjL1,axiom,
! [Xx: $i,Xy: $i,Xz: $i] :
( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
=> ( Xx = Xz ) ) ).
thf(kfstsingleton,axiom,
! [Xu: $i] :
( ( iskpair @ Xu )
=> ( singleton
@ ( dsetconstr @ ( setunion @ Xu )
@ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ).
thf(theprop,axiom,
! [X: $i] :
( ( singleton @ X )
=> ( in @ ( setunion @ X ) @ X ) ) ).
thf(kfst_type,type,
kfst: $i > $i ).
thf(kfstpairEq,axiom,
! [Xx: $i,Xy: $i] :
( ( kfst @ ( kpair @ Xx @ Xy ) )
= Xx ) ).
thf(cartprodfstin,axiom,
! [A: $i,B: $i,Xu: $i] :
( ( in @ Xu @ ( cartprod @ A @ B ) )
=> ( in @ ( kfst @ Xu ) @ A ) ) ).
thf(setukpairinjL2,axiom,
! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
=> ( Xx = Xz ) ) ).
thf(setukpairinjL,axiom,
! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
( ( ( kpair @ Xx @ Xy )
= ( kpair @ Xz @ Xu ) )
=> ( Xx = Xz ) ) ).
thf(setukpairinjR11,axiom,
! [Xx: $i,Xy: $i] :
( ( Xx = Xy )
=> ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
= ( setadjoin @ Xx @ emptyset ) ) ) ).
thf(setukpairinjR12,axiom,
! [Xx: $i,Xy: $i] :
( ( Xx = Xy )
=> ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ).
thf(setukpairinjR1,axiom,
! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
=> ( ( Xz = Xu )
=> ( Xy = Xu ) ) ) ).
thf(upairequniteq,axiom,
! [Xx: $i,Xy: $i,Xz: $i] :
( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
= ( setadjoin @ Xz @ emptyset ) )
=> ( Xx = Xy ) ) ).
thf(setukpairinjR2,axiom,
! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
=> ( Xy = Xu ) ) ).
thf(setukpairinjR,axiom,
! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
( ( ( kpair @ Xx @ Xy )
= ( kpair @ Xz @ Xu ) )
=> ( Xy = Xu ) ) ).
thf(ksndsingleton,axiom,
! [Xu: $i] :
( ( iskpair @ Xu )
=> ( singleton
@ ( dsetconstr @ ( setunion @ Xu )
@ ^ [Xx: $i] :
( Xu
= ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ).
thf(ksnd_type,type,
ksnd: $i > $i ).
thf(ksndpairEq,axiom,
! [Xx: $i,Xy: $i] :
( ( ksnd @ ( kpair @ Xx @ Xy ) )
= Xy ) ).
thf(kpairsurjEq,axiom,
! [Xu: $i] :
( ( iskpair @ Xu )
=> ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
= Xu ) ) ).
thf(cartprodsndin,axiom,
! [A: $i,B: $i,Xu: $i] :
( ( in @ Xu @ ( cartprod @ A @ B ) )
=> ( in @ ( ksnd @ Xu ) @ B ) ) ).
thf(cartprodpairmemEL,axiom,
! [A: $i,B: $i,Xx: $i,Xy: $i] :
( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
=> ( in @ Xx @ A ) ) ).
thf(cartprodpairmemER,axiom,
! [A: $i,B: $i,Xx: $i,Xy: $i] :
( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
=> ( in @ Xy @ B ) ) ).
thf(cartprodmempaircEq,axiom,
! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( ( kpair @ Xx @ Xy )
= ( kpair @ Xx @ Xy ) ) ) ) ).
thf(cartprodfstpairEq,axiom,
! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( ( kfst @ ( kpair @ Xx @ Xy ) )
= Xx ) ) ) ).
thf(cartprodsndpairEq,axiom,
! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( ( ksnd @ ( kpair @ Xx @ Xy ) )
= Xy ) ) ) ).
thf(cartprodpairsurjEq,axiom,
! [A: $i,B: $i,Xu: $i] :
( ( in @ Xu @ ( cartprod @ A @ B ) )
=> ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
= Xu ) ) ).
thf(breln_type,type,
breln: $i > $i > $i > $o ).
thf(dpsetconstr_type,type,
dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
thf(dpsetconstrI,axiom,
! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( ( Xphi @ Xx @ Xy )
=> ( in @ ( kpair @ Xx @ Xy )
@ ( dpsetconstr @ A @ B
@ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ).
thf(dpsetconstrSub,axiom,
! [A: $i,B: $i,Xphi: $i > $i > $o] :
( subset
@ ( dpsetconstr @ A @ B
@ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
@ ( cartprod @ A @ B ) ) ).
thf(setOfPairsIsBReln,axiom,
! [A: $i,B: $i,Xphi: $i > $i > $o] :
( breln @ A @ B
@ ( dpsetconstr @ A @ B
@ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ).
thf(dpsetconstrERa,axiom,
! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( ( in @ ( kpair @ Xx @ Xy )
@ ( dpsetconstr @ A @ B
@ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
=> ( Xphi @ Xx @ Xy ) ) ) ) ).
thf(dpsetconstrEL1,axiom,
! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
( ( in @ ( kpair @ Xx @ Xy )
@ ( dpsetconstr @ A @ B
@ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
=> ( in @ Xx @ A ) ) ).
thf(dpsetconstrEL2,axiom,
! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
( ( in @ ( kpair @ Xx @ Xy )
@ ( dpsetconstr @ A @ B
@ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
=> ( in @ Xy @ B ) ) ).
thf(dpsetconstrER,axiom,
! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
( ( in @ ( kpair @ Xx @ Xy )
@ ( dpsetconstr @ A @ B
@ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
=> ( Xphi @ Xx @ Xy ) ) ).
thf(func_type,type,
func: $i > $i > $i > $o ).
thf(funcSet_type,type,
funcSet: $i > $i > $i ).
thf(funcImageSingleton,axiom,
! [A: $i,B: $i,Xf: $i] :
( ( func @ A @ B @ Xf )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( singleton
@ ( dsetconstr @ B
@ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ).
thf(apProp,axiom,
! [A: $i,B: $i,Xf: $i] :
( ( func @ A @ B @ Xf )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( in
@ ( setunion
@ ( dsetconstr @ B
@ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
@ B ) ) ) ).
thf(ap_type,type,
ap: $i > $i > $i > $i > $i ).
thf(app,axiom,
! [A: $i,B: $i,Xf: $i] :
( ( func @ A @ B @ Xf )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ).
thf(infuncsetfunc,axiom,
! [A: $i,B: $i,Xf: $i] :
( ( in @ Xf @ ( funcSet @ A @ B ) )
=> ( func @ A @ B @ Xf ) ) ).
thf(ap2p,axiom,
! [A: $i,B: $i,Xf: $i] :
( ( in @ Xf @ ( funcSet @ A @ B ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ).
thf(funcinfuncset,axiom,
! [A: $i,B: $i,Xf: $i] :
( ( func @ A @ B @ Xf )
=> ( in @ Xf @ ( funcSet @ A @ B ) ) ) ).
thf(lamProp,axiom,
! [A: $i,B: $i,Xf: $i > $i] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ ( Xf @ Xx ) @ B ) )
=> ( func @ A @ B
@ ( dpsetconstr @ A @ B
@ ^ [Xx: $i,Xy: $i] :
( ( Xf @ Xx )
= Xy ) ) ) ) ).
thf(lam_type,type,
lam: $i > $i > ( $i > $i ) > $i ).
thf(lamp,axiom,
! [A: $i,B: $i,Xf: $i > $i] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ ( Xf @ Xx ) @ B ) )
=> ( func @ A @ B
@ ( lam @ A @ B
@ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ).
thf(lam2p,axiom,
! [A: $i,B: $i,Xf: $i > $i] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ ( Xf @ Xx ) @ B ) )
=> ( in
@ ( lam @ A @ B
@ ^ [Xx: $i] : ( Xf @ Xx ) )
@ ( funcSet @ A @ B ) ) ) ).
thf(brelnall1,axiom,
! [A: $i,B: $i,R: $i] :
( ( breln @ A @ B @ R )
=> ! [Xphi: $i > $o] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ R )
=> ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
=> ! [Xx: $i] :
( ( in @ Xx @ R )
=> ( Xphi @ Xx ) ) ) ) ).
thf(brelnall2,axiom,
! [A: $i,B: $i,R: $i] :
( ( breln @ A @ B @ R )
=> ! [Xphi: $i > $o] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ R )
=> ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
=> ! [Xx: $i] :
( ( in @ Xx @ R )
=> ( Xphi @ Xx ) ) ) ) ).
thf(ex1E2,axiom,
! [A: $i,Xphi: $i > $o] :
( ( ex1 @ A
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ( Xphi @ Xx )
=> ( ( Xphi @ Xy )
=> ( Xx = Xy ) ) ) ) ) ) ).
thf(funcGraphProp1,axiom,
! [A: $i,B: $i,Xf: $i] :
( ( func @ A @ B @ Xf )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ).
thf(funcGraphProp3,axiom,
! [A: $i,B: $i,Xf: $i] :
( ( in @ Xf @ ( funcSet @ A @ B ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ).
thf(funcGraphProp2,axiom,
! [A: $i,B: $i,Xf: $i] :
( ( func @ A @ B @ Xf )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
=> ( ( ap @ A @ B @ Xf @ Xx )
= Xy ) ) ) ) ) ).
thf(funcextLem,axiom,
! [A: $i,B: $i,Xf: $i] :
( ( func @ A @ B @ Xf )
=> ! [Xg: $i] :
( ( func @ A @ B @ Xg )
=> ( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( ap @ A @ B @ Xf @ Xx )
= ( ap @ A @ B @ Xg @ Xx ) ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
=> ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
thf(funcGraphProp4,axiom,
! [A: $i,B: $i,Xf: $i] :
( ( in @ Xf @ ( funcSet @ A @ B ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
=> ( ( ap @ A @ B @ Xf @ Xx )
= Xy ) ) ) ) ) ).
thf(subbreln,axiom,
! [A: $i,B: $i,R: $i] :
( ( breln @ A @ B @ R )
=> ! [S: $i] :
( ( breln @ A @ B @ S )
=> ( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ R )
=> ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
=> ( subset @ R @ S ) ) ) ) ).
thf(eqbreln,axiom,
! [A: $i,B: $i,R: $i] :
( ( breln @ A @ B @ R )
=> ! [S: $i] :
( ( breln @ A @ B @ S )
=> ( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ R )
=> ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
=> ( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ S )
=> ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
=> ( R = S ) ) ) ) ) ).
thf(funcext,axiom,
! [A: $i,B: $i,Xf: $i] :
( ( func @ A @ B @ Xf )
=> ! [Xg: $i] :
( ( func @ A @ B @ Xg )
=> ( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( ap @ A @ B @ Xf @ Xx )
= ( ap @ A @ B @ Xg @ Xx ) ) )
=> ( Xf = Xg ) ) ) ) ).
thf(funcext2,axiom,
! [A: $i,B: $i,Xf: $i] :
( ( in @ Xf @ ( funcSet @ A @ B ) )
=> ! [Xg: $i] :
( ( in @ Xg @ ( funcSet @ A @ B ) )
=> ( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( ap @ A @ B @ Xf @ Xx )
= ( ap @ A @ B @ Xg @ Xx ) ) )
=> ( Xf = Xg ) ) ) ) ).
thf(ap2apEq1,axiom,
! [A: $i,B: $i,Xf: $i] :
( ( in @ Xf @ ( funcSet @ A @ B ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( ap @ A @ B @ Xf @ Xx )
= ( ap @ A @ B @ Xf @ Xx ) ) ) ) ).
thf(ap2apEq2,axiom,
! [A: $i,B: $i,Xf: $i] :
( ( func @ A @ B @ Xf )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( ap @ A @ B @ Xf @ Xx )
= ( ap @ A @ B @ Xf @ Xx ) ) ) ) ).
thf(beta1,axiom,
! [A: $i,B: $i,Xf: $i > $i] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ ( Xf @ Xx ) @ B ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( ap @ A @ B
@ ( lam @ A @ B
@ ^ [Xy: $i] : ( Xf @ Xy ) )
@ Xx )
= ( Xf @ Xx ) ) ) ) ).
thf(eta1,axiom,
! [A: $i,B: $i,Xf: $i] :
( ( func @ A @ B @ Xf )
=> ( ( lam @ A @ B
@ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
= Xf ) ) ).
thf(lam2lamEq,axiom,
! [A: $i,B: $i,Xf: $i > $i] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ ( Xf @ Xx ) @ B ) )
=> ( ( lam @ A @ B
@ ^ [Xx: $i] : ( Xf @ Xx ) )
= ( lam @ A @ B
@ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ).
thf(beta2,axiom,
! [A: $i,B: $i,Xf: $i > $i] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ ( Xf @ Xx ) @ B ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( ap @ A @ B
@ ( lam @ A @ B
@ ^ [Xy: $i] : ( Xf @ Xy ) )
@ Xx )
= ( Xf @ Xx ) ) ) ) ).
thf(eta2,axiom,
! [A: $i,B: $i,Xf: $i] :
( ( in @ Xf @ ( funcSet @ A @ B ) )
=> ( ( lam @ A @ B
@ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
= Xf ) ) ).
thf(iffalseProp1,axiom,
! [A: $i,Xphi: $o,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ~ Xphi
=> ( in @ Xy
@ ( dsetconstr @ A
@ ^ [Xz: $i] :
( ( Xphi
& ( Xz = Xx ) )
| ( ~ Xphi
& ( Xz = Xy ) ) ) ) ) ) ) ) ).
thf(iffalseProp2,axiom,
! [A: $i,Xphi: $o,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ~ Xphi
=> ( ( dsetconstr @ A
@ ^ [Xz: $i] :
( ( Xphi
& ( Xz = Xx ) )
| ( ~ Xphi
& ( Xz = Xy ) ) ) )
= ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
thf(iftrueProp1,axiom,
! [A: $i,Xphi: $o,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( Xphi
=> ( in @ Xx
@ ( dsetconstr @ A
@ ^ [Xz: $i] :
( ( Xphi
& ( Xz = Xx ) )
| ( ~ Xphi
& ( Xz = Xy ) ) ) ) ) ) ) ) ).
thf(iftrueProp2,axiom,
! [A: $i,Xphi: $o,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( Xphi
=> ( ( dsetconstr @ A
@ ^ [Xz: $i] :
( ( Xphi
& ( Xz = Xx ) )
| ( ~ Xphi
& ( Xz = Xy ) ) ) )
= ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
thf(ifSingleton,axiom,
! [A: $i,Xphi: $o,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( singleton
@ ( dsetconstr @ A
@ ^ [Xz: $i] :
( ( Xphi
& ( Xz = Xx ) )
| ( ~ Xphi
& ( Xz = Xy ) ) ) ) ) ) ) ).
thf(if_type,type,
if: $i > $o > $i > $i > $i ).
thf(ifp,axiom,
! [A: $i,Xphi: $o,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ).
thf(theeq,axiom,
! [X: $i] :
( ( singleton @ X )
=> ! [Xx: $i] :
( ( in @ Xx @ X )
=> ( ( setunion @ X )
= Xx ) ) ) ).
thf(iftrue,axiom,
! [A: $i,Xphi: $o,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( Xphi
=> ( ( if @ A @ Xphi @ Xx @ Xy )
= Xx ) ) ) ) ).
thf(iffalse,axiom,
! [A: $i,Xphi: $o,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ~ Xphi
=> ( ( if @ A @ Xphi @ Xx @ Xy )
= Xy ) ) ) ) ).
thf(iftrueorfalse,axiom,
! [A: $i,Xphi: $o,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
thf(binintersectT_lem,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ).
thf(binunionT_lem,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ).
thf(powersetT_lem,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ).
thf(setminusT_lem,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ).
thf(complementT_lem,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ).
thf(setextT,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ X )
=> ( in @ Xx @ Y ) ) )
=> ( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ Y )
=> ( in @ Xx @ X ) ) )
=> ( X = Y ) ) ) ) ) ).
thf(subsetTI,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ X )
=> ( in @ Xx @ Y ) ) )
=> ( subset @ X @ Y ) ) ) ) ).
thf(powersetTI1,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ X )
=> ( in @ Xx @ Y ) ) )
=> ( in @ X @ ( powerset @ Y ) ) ) ) ) ).
thf(powersetTE1,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ X @ ( powerset @ Y ) )
=> ( ( in @ Xx @ X )
=> ( in @ Xx @ Y ) ) ) ) ) ) ).
thf(complementTI1,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ X )
=> ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ).
thf(complementTE1,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
=> ( in @ Xx @ X ) ) ) ) ).
thf(binintersectTELcontra,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ~ ( in @ Xx @ X )
=> ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ).
thf(binintersectTERcontra,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ~ ( in @ Xx @ Y )
=> ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ).
thf(contrasubsetT,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( subset @ X @ ( setminus @ A @ Y ) )
=> ( ( in @ Xx @ Y )
=> ~ ( in @ Xx @ X ) ) ) ) ) ) ).
thf(contrasubsetT1,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( subset @ X @ Y )
=> ( ~ ( in @ Xx @ Y )
=> ~ ( in @ Xx @ X ) ) ) ) ) ) ).
thf(contrasubsetT2,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( ( subset @ X @ Y )
=> ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ).
thf(contrasubsetT3,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
=> ( subset @ X @ Y ) ) ) ) ).
thf(doubleComplementI1,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ X )
=> ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
thf(doubleComplementE1,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
=> ( in @ Xx @ X ) ) ) ) ).
thf(doubleComplementSub1,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ).
thf(doubleComplementSub2,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ).
thf(doubleComplementEq,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ( X
= ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ).
thf(complementTnotintersectT,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ ( setminus @ A @ X ) )
=> ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ).
thf(complementImpComplementIntersect,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ ( setminus @ A @ X ) )
=> ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
thf(complementSubsetComplementIntersect,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ).
thf(complementInPowersetComplementIntersect,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ).
thf(contraSubsetComplement,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( ( subset @ X @ ( setminus @ A @ Y ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ Y )
=> ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
thf(complementTcontraSubset,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( ( subset @ X @ ( setminus @ A @ Y ) )
=> ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ).
thf(binunionTILcontra,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
=> ~ ( in @ Xx @ X ) ) ) ) ) ).
thf(binunionTIRcontra,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
=> ~ ( in @ Xx @ Y ) ) ) ) ) ).
thf(inIntersectImpInUnion,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Z: $i] :
( ( in @ Z @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ ( binintersect @ X @ Y ) )
=> ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ).
thf(inIntersectImpInUnion2,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Z: $i] :
( ( in @ Z @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ ( binintersect @ X @ Y ) )
=> ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ).
thf(inIntersectImpInIntersectUnions,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Z: $i] :
( ( in @ Z @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ ( binintersect @ X @ Y ) )
=> ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ).
thf(intersectInPowersetIntersectUnions,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Z: $i] :
( ( in @ Z @ ( powerset @ A ) )
=> ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ).
thf(inComplementUnionImpNotIn1,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
=> ~ ( in @ Xx @ X ) ) ) ) ) ).
thf(inComplementUnionImpInComplement1,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
=> ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ).
thf(binunionTE,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xphi: $o,Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ ( binunion @ X @ Y ) )
=> ( ( ( in @ Xx @ X )
=> Xphi )
=> ( ( ( in @ Xx @ Y )
=> Xphi )
=> Xphi ) ) ) ) ) ) ).
thf(binunionTEcontra,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ~ ( in @ Xx @ X )
=> ( ~ ( in @ Xx @ Y )
=> ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ).
thf(demorgan2a1,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
=> ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ).
thf(complementUnionInPowersetComplement,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ).
thf(demorgan2a2,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
=> ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ).
thf(demorgan1a,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
=> ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
thf(demorgan1b,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
=> ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
thf(demorgan1,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( ( setminus @ A @ ( binintersect @ X @ Y ) )
= ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ).
thf(demorgan2a,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
=> ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
thf(demorgan2b2,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ ( setminus @ A @ X ) )
=> ( ( in @ Xx @ ( setminus @ A @ Y ) )
=> ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ).
thf(demorgan2b,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
=> ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ).
thf(demorgan2,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( ( setminus @ A @ ( binunion @ X @ Y ) )
= ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ).
thf(woz13rule0,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ ( binintersect @ X @ Y ) )
=> ( in @ Xx @ A ) ) ) ) ).
thf(woz13rule1,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Z: $i] :
( ( in @ Z @ ( powerset @ A ) )
=> ( ( subset @ X @ Z )
=> ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ).
thf(woz13rule2,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Z: $i] :
( ( in @ Z @ ( powerset @ A ) )
=> ( ( subset @ Y @ Z )
=> ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ).
thf(woz13rule3,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Z: $i] :
( ( in @ Z @ ( powerset @ A ) )
=> ( ( subset @ X @ Y )
=> ( ( subset @ X @ Z )
=> ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ).
thf(woz13rule4,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Z: $i] :
( ( in @ Z @ ( powerset @ A ) )
=> ! [W: $i] :
( ( in @ W @ ( powerset @ A ) )
=> ( ( subset @ X @ Z )
=> ( ( subset @ Y @ W )
=> ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ).
thf(woz1_1,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ).
thf(woz1_2,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Z: $i] :
( ( in @ Z @ ( powerset @ A ) )
=> ! [W: $i] :
( ( in @ W @ ( powerset @ A ) )
=> ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
= ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ).
thf(woz1_3,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Z: $i] :
( ( in @ Z @ ( powerset @ A ) )
=> ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ).
thf(woz1_4,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( ( subset @ X @ ( setminus @ A @ Y ) )
=> ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ).
thf(woz1_5,axiom,
! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ).
thf(breln1_type,type,
breln1: $i > $i > $o ).
thf(breln1all2,axiom,
! [A: $i,R: $i] :
( ( breln1 @ A @ R )
=> ! [Xphi: $i > $o] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ R )
=> ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
=> ! [Xx: $i] :
( ( in @ Xx @ R )
=> ( Xphi @ Xx ) ) ) ) ).
thf(breln1Set_type,type,
breln1Set: $i > $i ).
thf(breln1SetBreln1,axiom,
! [A: $i,R: $i] :
( ( in @ R @ ( breln1Set @ A ) )
=> ( breln1 @ A @ R ) ) ).
thf(transitive_type,type,
transitive: $i > $i > $o ).
thf(antisymmetric_type,type,
antisymmetric: $i > $i > $o ).
thf(reflexive_type,type,
reflexive: $i > $i > $o ).
thf(refltransitive_type,type,
refltransitive: $i > $i > $o ).
thf(refllinearorder_type,type,
refllinearorder: $i > $i > $o ).
thf(reflwellordering_type,type,
reflwellordering: $i > $i > $o ).
thf(choice2fnsingleton,axiom,
! [A: $i,B: $i,Xphi: $i > $i > $o] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ? [Xy: $i] :
( ( in @ Xy @ B )
& ( Xphi @ Xx @ Xy ) ) )
=> ! [R: $i] :
( ( in @ R @ ( breln1Set @ B ) )
=> ( ( reflwellordering @ B @ R )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( singleton
@ ( dsetconstr @ B
@ ^ [Xy: $i] :
( ( Xphi @ Xx @ Xy )
& ! [Xz: $i] :
( ( in @ Xz @ B )
=> ( ( Xphi @ Xx @ Xz )
=> ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ).
thf(setOfPairsIsBReln1,axiom,
! [A: $i,Xphi: $i > $i > $o] :
( breln1 @ A
@ ( dpsetconstr @ A @ A
@ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ).
thf(breln1all1,axiom,
! [A: $i,R: $i] :
( ( breln1 @ A @ R )
=> ! [Xphi: $i > $o] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ R )
=> ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
=> ! [Xx: $i] :
( ( in @ Xx @ R )
=> ( Xphi @ Xx ) ) ) ) ).
thf(subbreln1,axiom,
! [A: $i,R: $i] :
( ( breln1 @ A @ R )
=> ! [S: $i] :
( ( breln1 @ A @ S )
=> ( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ R )
=> ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
=> ( subset @ R @ S ) ) ) ) ).
thf(eqbreln1,axiom,
! [A: $i,R: $i] :
( ( breln1 @ A @ R )
=> ! [S: $i] :
( ( breln1 @ A @ S )
=> ( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ R )
=> ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
=> ( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ S )
=> ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
=> ( R = S ) ) ) ) ) ).
thf(breln1invset_type,type,
breln1invset: $i > $i > $i ).
thf(breln1invprop,axiom,
! [A: $i,R: $i] :
( ( breln1 @ A @ R )
=> ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ).
thf(breln1invI,axiom,
! [A: $i,R: $i] :
( ( breln1 @ A @ R )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ R )
=> ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ).
thf(breln1invE,axiom,
! [A: $i,R: $i] :
( ( breln1 @ A @ R )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) )
=> ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ).
thf(breln1compset_type,type,
breln1compset: $i > $i > $i > $i ).
thf(breln1compprop,axiom,
! [A: $i,R: $i] :
( ( breln1 @ A @ R )
=> ! [S: $i] :
( ( breln1 @ A @ S )
=> ( breln1 @ A @ ( breln1compset @ A @ R @ S ) ) ) ) ).
thf(breln1compI,axiom,
! [A: $i,R: $i] :
( ( breln1 @ A @ R )
=> ! [S: $i] :
( ( breln1 @ A @ S )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ! [Xz: $i] :
( ( in @ Xz @ A )
=> ( ( in @ ( kpair @ Xx @ Xz ) @ R )
=> ( ( in @ ( kpair @ Xz @ Xy ) @ S )
=> ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ) ) ) ).
thf(breln1compE,axiom,
! [A: $i,R: $i] :
( ( breln1 @ A @ R )
=> ! [S: $i] :
( ( breln1 @ A @ S )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
=> ? [Xz: $i] :
( ( in @ Xz @ A )
& ( in @ ( kpair @ Xx @ Xz ) @ R )
& ( in @ ( kpair @ Xz @ Xy ) @ S ) ) ) ) ) ) ) ).
thf(breln1compEex,axiom,
! [A: $i,R: $i] :
( ( breln1 @ A @ R )
=> ! [S: $i] :
( ( breln1 @ A @ S )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
=> ! [Xphi: $o] :
( ! [Xz: $i] :
( ( in @ Xz @ A )
=> ( ( in @ ( kpair @ Xx @ Xz ) @ R )
=> ( ( in @ ( kpair @ Xz @ Xy ) @ S )
=> Xphi ) ) )
=> Xphi ) ) ) ) ) ) ).
thf(breln1unionprop,axiom,
! [A: $i,R: $i] :
( ( breln1 @ A @ R )
=> ! [S: $i] :
( ( breln1 @ A @ S )
=> ( breln1 @ A @ ( binunion @ R @ S ) ) ) ) ).
thf(breln1unionIL,axiom,
! [A: $i,R: $i] :
( ( breln1 @ A @ R )
=> ! [S: $i] :
( ( breln1 @ A @ S )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ R )
=> ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ).
thf(breln1unionIR,axiom,
! [A: $i,R: $i] :
( ( breln1 @ A @ R )
=> ! [S: $i] :
( ( breln1 @ A @ S )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ S )
=> ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ).
thf(breln1unionI,axiom,
! [A: $i,R: $i] :
( ( breln1 @ A @ R )
=> ! [S: $i] :
( ( breln1 @ A @ S )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
| ( in @ ( kpair @ Xx @ Xy ) @ S ) )
=> ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ).
thf(breln1unionE,axiom,
! [A: $i,R: $i] :
( ( breln1 @ A @ R )
=> ! [S: $i] :
( ( breln1 @ A @ S )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ R )
| ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) ) ) ) ) ).
thf(breln1unionEcases,axiom,
! [A: $i,R: $i] :
( ( breln1 @ A @ R )
=> ! [S: $i] :
( ( breln1 @ A @ S )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
=> ! [Xphi: $o] :
( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
=> Xphi )
=> ( ( ( in @ ( kpair @ Xx @ Xy ) @ S )
=> Xphi )
=> Xphi ) ) ) ) ) ) ) ).
thf(breln1unionCommutes,axiom,
! [A: $i,R: $i] :
( ( breln1 @ A @ R )
=> ! [S: $i] :
( ( breln1 @ A @ S )
=> ( ( binunion @ R @ S )
= ( binunion @ S @ R ) ) ) ) ).
thf(woz2Ex,axiom,
! [A: $i,R: $i] :
( ( breln1 @ A @ R )
=> ( R
= ( breln1invset @ A @ ( breln1invset @ A @ R ) ) ) ) ).
thf(woz2W,axiom,
! [A: $i,R: $i] :
( ( breln1 @ A @ R )
=> ! [S: $i] :
( ( breln1 @ A @ S )
=> ( ( breln1invset @ A @ ( breln1compset @ A @ R @ S ) )
= ( breln1compset @ A @ ( breln1invset @ A @ S ) @ ( breln1invset @ A @ R ) ) ) ) ) ).
thf(woz2A,axiom,
! [A: $i,R: $i] :
( ( breln1 @ A @ R )
=> ! [S: $i] :
( ( breln1 @ A @ S )
=> ! [T: $i] :
( ( breln1 @ A @ T )
=> ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
= ( binunion @ ( breln1compset @ A @ R @ T ) @ ( breln1compset @ A @ S @ T ) ) ) ) ) ) ).
thf(woz2B,axiom,
! [A: $i,R: $i] :
( ( breln1 @ A @ R )
=> ! [S: $i] :
( ( breln1 @ A @ S )
=> ! [T: $i] :
( ( breln1 @ A @ T )
=> ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
= ( binunion @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ S ) ) ) @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ).
thf(image1Ex,axiom,
! [A: $i,Xf: $i > $i] :
? [B: $i] :
! [Xx: $i] :
( ( in @ Xx @ B )
<=> ? [Xy: $i] :
( ( in @ Xy @ A )
& ( Xx
= ( Xf @ Xy ) ) ) ) ).
thf(image1Ex1,axiom,
! [A: $i,Xf: $i > $i] :
( exu
@ ^ [B: $i] :
! [Xx: $i] :
( ( in @ Xx @ B )
<=> ? [Xy: $i] :
( ( in @ Xy @ A )
& ( Xx
= ( Xf @ Xy ) ) ) ) ) ).
thf(image1_type,type,
image1: $i > ( $i > $i ) > $i ).
thf(image1Equiv,axiom,
! [A: $i,Xf: $i > $i,Xx: $i] :
( ( in @ Xx
@ ( image1 @ A
@ ^ [Xy: $i] : ( Xf @ Xy ) ) )
<=> ? [Xy: $i] :
( ( in @ Xy @ A )
& ( Xx
= ( Xf @ Xy ) ) ) ) ).
thf(image1E,axiom,
! [A: $i,Xf: $i > $i,Xx: $i] :
( ( in @ Xx
@ ( image1 @ A
@ ^ [Xy: $i] : ( Xf @ Xy ) ) )
=> ? [Xy: $i] :
( ( in @ Xy @ A )
& ( Xx
= ( Xf @ Xy ) ) ) ) ).
thf(image1I,axiom,
! [A: $i,Xf: $i > $i,Xx: $i] :
( ? [Xy: $i] :
( ( in @ Xy @ A )
& ( Xx
= ( Xf @ Xy ) ) )
=> ( in @ Xx
@ ( image1 @ A
@ ^ [Xy: $i] : ( Xf @ Xy ) ) ) ) ).
thf(injective_type,type,
injective: $i > $i > $i > $o ).
thf(injFuncSet_type,type,
injFuncSet: $i > $i > $i ).
thf(injFuncInInjFuncSet,axiom,
! [A: $i,B: $i,Xf: $i] :
( ( in @ Xf @ ( funcSet @ A @ B ) )
=> ( ( injective @ A @ B @ Xf )
=> ( in @ Xf @ ( injFuncSet @ A @ B ) ) ) ) ).
thf(injFuncSetFuncIn,axiom,
! [A: $i,B: $i,Xf: $i] :
( ( in @ Xf @ ( injFuncSet @ A @ B ) )
=> ( in @ Xf @ ( funcSet @ A @ B ) ) ) ).
thf(injFuncSetFuncInj,axiom,
! [Xx: $i,Xy: $i,Xf: $i] :
( ( in @ Xf @ ( injFuncSet @ Xx @ Xy ) )
=> ( injective @ Xx @ Xy @ Xf ) ) ).
thf(surjective_type,type,
surjective: $i > $i > $i > $o ).
thf(surjFuncSet_type,type,
surjFuncSet: $i > $i > $i ).
thf(surjFuncSetFuncIn,axiom,
! [A: $i,B: $i,Xf: $i] :
( ( in @ Xf @ ( surjFuncSet @ A @ B ) )
=> ( in @ Xf @ ( funcSet @ A @ B ) ) ) ).
thf(surjFuncSetFuncSurj,axiom,
! [Xx: $i,Xy: $i,Xf: $i] :
( ( in @ Xf @ ( surjFuncSet @ Xx @ Xy ) )
=> ( surjective @ Xx @ Xy @ Xf ) ) ).
thf(leftInvIsSurj,axiom,
! [A: $i,B: $i,Xf: $i > $i] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ ( Xf @ Xx ) @ B ) )
=> ! [Xg: $i] :
( ( in @ Xg @ ( funcSet @ B @ A ) )
=> ( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( ap @ B @ A @ Xg @ ( Xf @ Xx ) )
= Xx ) )
=> ( surjective @ B @ A @ Xg ) ) ) ) ).
thf(surjCantorThm,axiom,
! [A: $i,Xf: $i] :
( ( in @ Xf @ ( funcSet @ A @ ( powerset @ A ) ) )
=> ~ ( surjective @ A @ ( powerset @ A ) @ Xf ) ) ).
thf(foundation2,axiom,
! [A: $i] :
( ( nonempty @ A )
=> ? [X: $i] :
( ( in @ X @ A )
& ( ( binintersect @ X @ A )
= emptyset ) ) ) ).
thf(notinself,axiom,
! [A: $i] :
~ ( in @ A @ A ) ).
thf(notinself2,axiom,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ~ ( in @ B @ A ) ) ).
thf(omegaS_type,type,
omegaS: $i > $i ).
thf(omegaSp,axiom,
! [Xx: $i] :
( ( in @ Xx @ omega )
=> ( in @ ( omegaS @ Xx ) @ omega ) ) ).
thf(omegaSclos,axiom,
! [Xx: $i] :
( ( in @ Xx @ omega )
=> ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ).
thf(peano0notS,axiom,
! [Xx: $i] :
( ( in @ Xx @ omega )
=> ( ( omegaS @ Xx )
!= emptyset ) ) ).
thf(peano3_type,type,
peano3: $i > $i > $i > $o ).
thf(peano4_type,type,
peano4: $i > $i > $i > $o ).
thf(peano5_type,type,
peano5: $i > $i > $i > $o ).
thf(peanoSinj,axiom,
! [Xx: $i] :
( ( in @ Xx @ omega )
=> ! [Xy: $i] :
( ( in @ Xy @ omega )
=> ( ( ( omegaS @ Xx )
= ( omegaS @ Xy ) )
=> ( Xx = Xy ) ) ) ) ).
thf(peano_type,type,
peano: $i > $i > $i > $o ).
thf(transitiveset_type,type,
transitiveset: $i > $o ).
thf(transitivesetOp1,axiom,
! [X: $i] :
( ( transitiveset @ X )
=> ! [A: $i] :
( ( in @ A @ X )
=> ( subset @ A @ X ) ) ) ).
thf(binintTransitive,axiom,
! [X: $i] :
( ( transitiveset @ X )
=> ! [Y: $i] :
( ( transitiveset @ Y )
=> ( transitiveset @ ( binintersect @ X @ Y ) ) ) ) ).
thf(transitivesetOp2,axiom,
! [X: $i] :
( ( transitiveset @ X )
=> ! [A: $i,Xx: $i] :
( ( in @ A @ X )
=> ( ( in @ Xx @ A )
=> ( in @ Xx @ X ) ) ) ) ).
thf(setunionTransitive,axiom,
! [X: $i] :
( ! [Xx: $i] :
( ( in @ Xx @ X )
=> ( transitiveset @ Xx ) )
=> ( transitiveset @ ( setunion @ X ) ) ) ).
thf(stricttotalorderedByIn_type,type,
stricttotalorderedByIn: $i > $o ).
thf(stricttotalorderedByIn,definition,
( stricttotalorderedByIn
= ( ^ [A: $i] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [X: $i] :
( ( in @ X @ A )
=> ! [Y: $i] :
( ( in @ Y @ A )
=> ( ( ( in @ Xx @ X )
& ( in @ X @ Y ) )
=> ( in @ Xx @ Y ) ) ) ) )
& ! [X: $i] :
( ( in @ X @ A )
=> ! [Y: $i] :
( ( in @ Y @ A )
=> ( ( X = Y )
| ( in @ X @ Y )
| ( in @ Y @ X ) ) ) )
& ! [X: $i] :
( ( in @ X @ A )
=> ~ ( in @ X @ X ) ) ) ) ) ).
thf(wellorderedByIn_type,type,
wellorderedByIn: $i > $o ).
thf(wellorderedByIn,definition,
( wellorderedByIn
= ( ^ [A: $i] :
( ( stricttotalorderedByIn @ A )
& ! [X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ( ( nonempty @ X )
=> ? [Xx: $i] :
( ( in @ Xx @ X )
& ! [Y: $i] :
( ( in @ Y @ X )
=> ( ( Xx = Y )
| ( in @ Xx @ Y ) ) ) ) ) ) ) ) ) ).
thf(ordinal_type,type,
ordinal: $i > $o ).
thf(ordinal,definition,
( ordinal
= ( ^ [Xx: $i] :
( ( transitiveset @ Xx )
& ( wellorderedByIn @ Xx ) ) ) ) ).
thf(limitOrdinal_type,type,
limitOrdinal: $i > $o ).
thf(ordinalMinLem1,axiom,
! [X: $i] :
( ( ordinal @ X )
=> ! [Y: $i] :
( ( ordinal @ Y )
=> ( transitiveset @ ( binintersect @ X @ Y ) ) ) ) ).
thf(ordinalTransSet,axiom,
! [X: $i] :
( ( ordinal @ X )
=> ! [Xx: $i,A: $i] :
( ( in @ A @ X )
=> ( ( in @ Xx @ A )
=> ( in @ Xx @ X ) ) ) ) ).
thf(ordinalTransSet1,axiom,
! [X: $i] :
( ( ordinal @ X )
=> ! [A: $i] :
( ( in @ A @ X )
=> ( subset @ A @ X ) ) ) ).
thf(setunionOrdinalLem1,axiom,
! [X: $i] :
( ! [Xx: $i] :
( ( in @ Xx @ X )
=> ( ordinal @ Xx ) )
=> ( transitiveset @ ( setunion @ X ) ) ) ).
thf(succOrdinal_type,type,
succOrdinal: $i > $o ).
thf(emptysetOrdinal,axiom,
ordinal @ emptyset ).
thf(ordinalIrrefl,axiom,
! [X: $i] :
( ( ordinal @ X )
=> ! [A: $i] :
( ( in @ A @ X )
=> ~ ( in @ A @ A ) ) ) ).
thf(ordinalIrrefl2,axiom,
! [X: $i] :
( ( ordinal @ X )
=> ~ ( in @ X @ X ) ) ).
thf(ordinalNoCycle,axiom,
! [X: $i] :
( ( ordinal @ X )
=> ! [A: $i] :
( ( in @ X @ A )
=> ~ ( in @ A @ X ) ) ) ).
thf(ordinalTransIn,conjecture,
$false ).
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